I am also questioning the "well, he pays bills, fixes the toilet, mows the lawn and so it all evens out" idea. If one partner is doing daily tasks and is responsible for all child-care related planning, and the other is doing only weekly or monthly or seasonal or ad hoc tasks and not taking responsibility for planning - that is not an even distribution of work.
I am also firmly against having sex if you are feeling tired, angry, and resentful of him. I am further against trading sex for household work. No.
Write down all the things you do on a daily/weekly/monthly/seasonal/ad hoc basis to keep the household going. Include scheduling the kid's appointments and activities. Be honest. Don't leave anything out.
Write down all the things he does on a daily/weekly/monthly/seasonal/ad hoc basis to keep the household going. Be honest. Don't leave anything out, but don't give him full credit for anything that YOU MUST ASK HIM TO DO FIRST. Like do you have to ask him to mow the lawn? Do you have to ask him to fix the toilet. If so, then responsibility for planning that goes on your jobs list.
Now, take a look at your lists. How many things on his list could you afford to hire someone to do if you were only going with your salary? Could you afford your house on your salary alone? If you had to hire someone to do his stuff too?
If you choose to divorce, it's not over chores, it's over lack of respect. If you choose not to divorce, make sure that the change comes not from him doing more, but from his valuing you and what you do.
And let him do his own damned laundry.

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